Though you may hear both terms in reference to DNA, genotyping and sequencing refer to slightly different techniques.
Genotyping is the process of determining which genetic variants an individual possesses. Genotyping can be performed through a variety of different methods, depending on the variants of interest and the resources available. For looking at many different variants at once, especially common variants, genotyping chips are an efficient and accurate method. They do, however, require prior identification of the variants of interest.
Sequencing is a method used to determine the exact sequence of a certain length of DNA. You can sequence a short piece, the whole genome, or parts of the genome such as the "exome," which are the regions of the genome that contain the instructions for RNAs (ribonucleic acid) and proteins. Depending on the region, a given stretch of sequence may include some DNA that varies between individuals, in addition to regions that are constant. Thus, sequencing can be used to genotype someone for known variants, as well as identify variants that may be unique to that person.
23andMe uses genotyping, not sequencing, to analyze your DNA. Sequencing technology has not yet progressed to the point where it is feasible to sequence an entire person’s genome quickly and cheaply enough to keep costs down for consumers. It took the Human Genome Project, a consortium of multiple research labs, over 10 years to sequence the whole genomes of just a few individuals.
For people with specific needs (undiagnosed medical issues, for example, or intense curiosity and a generous budget), sequencing will become a more and more attractive option. For now, though, genotyping technologies, such as those used by 23andMe, provide an efficient and cost-effective way of evaluating genetic variation in individuals and across populations.